Finding derivative of x^x using limit definition

In summary, the limit definition of a derivative is a formula used to find the instantaneous rate of change of a function at a specific point. It involves taking the limit of a difference quotient as the difference approaches zero. To find the derivative of x^x using this definition, we substitute the difference quotient into the formula and simplify until we have an expression without h. This method is important because it is the fundamental way of finding derivatives and allows us to calculate the instantaneous rate of change at any point on a function's graph. While this method can be used for any function, there are also other methods such as the power rule and logarithmic differentiation.
  • #1
scottshannon
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I am trying to find the derivative of x^x using the limit definition and am unable to follow what I have read. Can someone help me understand why lim [(x+h)^h -1]/h as h ---> 0 = ln(x). This part of the derivatio
 

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  • #2
Try using the fact that [itex]e^{ln(x)}=x[/itex]
 
  • #3
Limit (a∧h) = 1 + h×log(a)/1! + ((h×log(a))∧2)/2! + ...
h→0

For the steps performed on the right side , use , as already said , x = e∧log(x) ,
and

Limit (log(1+h)) = h - (h∧2)/2 + (h∧3)/3 - ...
h→0

*Log is taken to base e .
 

FAQ: Finding derivative of x^x using limit definition

1. What is the limit definition of a derivative?

The limit definition of a derivative is the mathematical formula used to find the instantaneous rate of change of a function at a specific point, also known as the slope of the tangent line. It is written as the limit of a difference quotient, where the difference in the function's output values is divided by the difference in the input values as the difference approaches zero.

2. How do you find the derivative of x^x using the limit definition?

To find the derivative of x^x using the limit definition, we first write the difference quotient as (f(x+h)-f(x))/h, where f(x)=x^x. Then, we substitute this into the limit definition formula and simplify until we are left with an expression that does not contain h. This final expression is the derivative of x^x.

3. Why is finding the derivative of x^x using the limit definition important?

Finding the derivative of x^x using the limit definition is important because it is the fundamental method for finding derivatives of any function. It allows us to calculate the instantaneous rate of change at any point on the graph of a function, which is crucial in many areas of science and engineering.

4. Can you find the derivative of x^x at any point?

Yes, the limit definition of a derivative allows us to find the derivative of x^x at any point. However, as the function x^x is continuous, the derivative may not exist at certain points. For example, the derivative at x=0 does not exist as x^x is undefined at that point.

5. Are there any other methods for finding the derivative of x^x?

Yes, there are other methods for finding the derivative of x^x, such as using the power rule or logarithmic differentiation. However, the limit definition is the most general method and can be applied to any function, making it a valuable tool in calculus and other fields of science.

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